Question
Yvette is considering taking out a loan with a principal of $16,200 from one of two banks. Bank F charges an interest rate of 5.7%, compounded monthly, and requires that the loan be paid off in eight years. Bank G charges an interest rate of 6.2%, compounded monthly, and requires that the loan be paid off in seven years. How would you recommend that Yvette choose her loan?
a.
Bank F offers a better loan in every regard, so Yvette should choose it over Bank G’s.
b.
Yvette should choose Bank F’s loan if she cares more about lower monthly payments, and she should choose Bank G’s loan if she cares more about the lowest lifetime cost.
c.
Yvette should choose Bank G’s loan if she cares more about lower monthly payments, and she should choose Bank F’s loan if she cares more about the lowest lifetime cost.
d.
Bank G offers a better loan in every regard, so Yvette should choose it over Bank F’s.
Please select the best answer from the choices provided
A
B
C
D
a.
Bank F offers a better loan in every regard, so Yvette should choose it over Bank G’s.
b.
Yvette should choose Bank F’s loan if she cares more about lower monthly payments, and she should choose Bank G’s loan if she cares more about the lowest lifetime cost.
c.
Yvette should choose Bank G’s loan if she cares more about lower monthly payments, and she should choose Bank F’s loan if she cares more about the lowest lifetime cost.
d.
Bank G offers a better loan in every regard, so Yvette should choose it over Bank F’s.
Please select the best answer from the choices provided
A
B
C
D
Answers
GPT-4o mini
To determine which loan is better for Yvette, we need to calculate the monthly payments and the total cost of each loan.
**For Bank F:**
- Principal (P) = $16,200
- Annual interest rate (r) = 5.7% = 0.057
- Monthly interest rate (i) = 0.057 / 12 = 0.00475
- Number of months (n) = 8 years × 12 months/year = 96 months
Using the formula for monthly payments:
\[
M = P \times \frac{i(1+i)^n}{(1+i)^n - 1}
\]
Substituting in the values:
\[
M = 16,200 \times \frac{0.00475(1 + 0.00475)^{96}}{(1 + 0.00475)^{96} - 1}
\]
Calculating the values:
1. \( (1 + 0.00475)^{96} \approx 1.432364654 \)
2. \( M = 16,200 \times \frac{0.00475 \times 1.432364654}{1.432364654 - 1} \approx 16,200 \times \frac{0.006843731}{0.432364654} \approx 254.32 \)
Total cost for Bank F:
\[
Total\ Cost = M \times n = 254.32 \times 96 = 24,375.72
\]
**For Bank G:**
- Principal (P) = $16,200
- Annual interest rate (r) = 6.2% = 0.062
- Monthly interest rate (i) = 0.062 / 12 = 0.00516667
- Number of months (n) = 7 years × 12 months/year = 84 months
Using the same payment formula:
\[
M = 16,200 \times \frac{0.00516667(1 + 0.00516667)^{84}}{(1 + 0.00516667)^{84} - 1}
\]
Calculating the values:
1. \( (1 + 0.00516667)^{84} \approx 1.486584395 \)
2. \( M = 16,200 \times \frac{0.00516667 \times 1.486584395}{1.486584395 - 1} \approx 16,200 \times \frac{0.007685983}{0.486584395} \approx 275.22 \)
Total cost for Bank G:
\[
Total\ Cost = M \times n = 275.22 \times 84 = 23,051.48
\]
**Comparison of Loans:**
- Total cost of loan from Bank F: $24,375.72
- Total cost of loan from Bank G: $23,051.48
- Monthly payment from Bank F: $254.32
- Monthly payment from Bank G: $275.22
**Conclusion:**
- Bank F has a lower monthly payment but a higher total cost.
- Bank G has a higher monthly payment but a lower total cost.
Considering the options:
- **Option B** states: "Yvette should choose Bank F’s loan if she cares more about lower monthly payments, and she should choose Bank G’s loan if she cares more about the lowest lifetime cost." This accurately reflects the analysis.
Thus, the best choice for Yvette is:
**B**
**For Bank F:**
- Principal (P) = $16,200
- Annual interest rate (r) = 5.7% = 0.057
- Monthly interest rate (i) = 0.057 / 12 = 0.00475
- Number of months (n) = 8 years × 12 months/year = 96 months
Using the formula for monthly payments:
\[
M = P \times \frac{i(1+i)^n}{(1+i)^n - 1}
\]
Substituting in the values:
\[
M = 16,200 \times \frac{0.00475(1 + 0.00475)^{96}}{(1 + 0.00475)^{96} - 1}
\]
Calculating the values:
1. \( (1 + 0.00475)^{96} \approx 1.432364654 \)
2. \( M = 16,200 \times \frac{0.00475 \times 1.432364654}{1.432364654 - 1} \approx 16,200 \times \frac{0.006843731}{0.432364654} \approx 254.32 \)
Total cost for Bank F:
\[
Total\ Cost = M \times n = 254.32 \times 96 = 24,375.72
\]
**For Bank G:**
- Principal (P) = $16,200
- Annual interest rate (r) = 6.2% = 0.062
- Monthly interest rate (i) = 0.062 / 12 = 0.00516667
- Number of months (n) = 7 years × 12 months/year = 84 months
Using the same payment formula:
\[
M = 16,200 \times \frac{0.00516667(1 + 0.00516667)^{84}}{(1 + 0.00516667)^{84} - 1}
\]
Calculating the values:
1. \( (1 + 0.00516667)^{84} \approx 1.486584395 \)
2. \( M = 16,200 \times \frac{0.00516667 \times 1.486584395}{1.486584395 - 1} \approx 16,200 \times \frac{0.007685983}{0.486584395} \approx 275.22 \)
Total cost for Bank G:
\[
Total\ Cost = M \times n = 275.22 \times 84 = 23,051.48
\]
**Comparison of Loans:**
- Total cost of loan from Bank F: $24,375.72
- Total cost of loan from Bank G: $23,051.48
- Monthly payment from Bank F: $254.32
- Monthly payment from Bank G: $275.22
**Conclusion:**
- Bank F has a lower monthly payment but a higher total cost.
- Bank G has a higher monthly payment but a lower total cost.
Considering the options:
- **Option B** states: "Yvette should choose Bank F’s loan if she cares more about lower monthly payments, and she should choose Bank G’s loan if she cares more about the lowest lifetime cost." This accurately reflects the analysis.
Thus, the best choice for Yvette is:
**B**